Wake characteristics in high solidity horizontal axis hydrokinetic turbines: a comparative study between experimental techniques and numerical simulations

Abstract

Hydrokinetic turbines utilizing renewable energy from water streams, such as tidal or river currents, offer a sustainable solution for electricity generation in remote coastal or fluvial regions. While there has been extensive research on the wake characteristics of low-solidity turbines, studies focusing on high-solidity turbines are limited. This study aims to address this research gap by characterizing the turbulent wake of a high-solidity, four-bladed hydrokinetic turbine using a combination of experimental and numerical approaches. Experimental investigations are conducted in two wind tunnel facilities, employing particle image velocimetry and hot-wire anemometry to measure and identify flow structures in the near-wake region. Additionally, the applicability of the actuator line method for high-solidity turbines is validated, considering the challenges posed by the higher blockage effect associated with such turbines. Numerical simulations are performed using an unsteady Reynolds-averaged Navier–Stokes approach, incorporating a full-geometry model of the rotating rotor and employing the actuator line concept for simplified modeling. The findings from this study provide valuable insights into the wake characteristics of high-solidity hydrokinetic turbines, contributing to the understanding of their performance and environmental implications.

Appendix

In this appendix, we describe the actuator line method [15] based on an iterative process, combining the blade element momentum (BEM) and CFD equations. The method solves the Navier–Stokes equations with a font term counting the hydrodynamic forces on the blades. Firstly, it computes the 2D hydrodynamic forces on the blades (${\textbf {f}}_{2d}$). Later, it projects these forces in the field flow using a regularization Kernel function ($\eta _{\epsilon }$) and incorporates the three-dimensional forces $f_i({\textbf {x}})$ in the Navier–Stokes equations.

In the ALM method, the rotor is a set of ‘points’ or ‘elements’ along the axis of the blade, as in the scheme in Fig. 15; it is no longer used as a surface, bringing advantages such as lower computational costs or the possibility of structured mesh, suitable for large-eddy simulations.

Fig. 15

Full rotor geometry and blade discretization scheme to ALM. The hydrodynamic drag and lift forces, D and L, referred to any section of the blade are illustrated

The steps and main parameters to implement the ALM are described as:

1.:

Rotor geometry definition: Rotor is constituted of 4 blades and 2.2 m in diameter, and the hydrodynamic profile is a NACA 4415. Other geometric data for the design of the blades (radial distribution, chord, and torsion angle) were extracted from the Hydrok project and are summarized in Table 2.

For the ALM, the blades were discretized in 38 points, an estimated value based on the relationship proposed by [44] such that $\varDelta x= R/n$, which relates the rotor radius, $R=1.1$ m, and the mesh element size, $\varDelta x=0.03$ m, for the mesh presented in this work. In this way, it guarantees an appropriate relation between mesh size and blade discretization to prevent more than one point reside in the same mesh element.

2.:

Turbine operating condition: characterized by free stream velocity $U_{0}=2.5$ m/s and rotational velocity $\varOmega =35$ rpm, and, consequently, defined by tip speed ratio $TSR=1.6$. After geometric and kinematic definition, Reynolds number is computed through the mean chord, $c_m$, and the relative flow velocity ($U_{rel}$) at mean radial position $R_m$, to operation conditions, such as $U_{rel}=\sqrt{U_{0}^2+(\varOmega R_m)^2}$. Thus, the computed local Reynolds number is $Re_L=2 \times 10^6$.

3.:

Hydrofoil polar curves: Using the free software XFOIL [45], which employs panel methods with a boundary layer formulation, the lift and drag coefficient, $C_L(\alpha ,Re_L)$ and $C_D(\alpha ,Re_L)$ as a function of angle of attack ($\alpha$) and local Reynolds number, are computed and presented in Fig. 16. The input values to the software are the 2D hydrofoil geometry, the local Reynolds numbers, and the Mach number.

4.:

Computing hydrodynamic forces by BEM method: The two-dimensional hydrodynamic forces per unit length to each blade element are defined as

$$\begin{aligned} {\textbf {f}}_{2d}= f_L {\textbf {e}}_L + f_D {\textbf {e}}_D, \end{aligned}$$

(4)

where $f_L$ and $f_D$ are calculated through equations 6 and 5, after to know the flow angle attack to each blade point, and to compute the relative flow velocity and the lift and drag coefficient.

$$\begin{aligned} f_D= & {} \frac{1}{2} C_D(\alpha ) \rho U_{rel}^2 c, \end{aligned}$$

(5)

$$\begin{aligned} f_L= & {} \frac{1}{2} C_L(\alpha ) \rho U_{rel}^2 c, \end{aligned}$$

(6)

where c is the chord length, $\rho$ the flow density, $U_{rel}$ the relative local velocity, and $C_L$ and $C_D$, the lift and drag coefficients, respectively.

In each radial position and angle of attack, there is a specific value of the relative local velocity as well as $C_L$ and $C_D$. So, the relative velocity is computed by

$$\begin{aligned} U_{rel}=\sqrt{(U_z^2+(\varOmega r-U_{\theta })^2)}, \end{aligned}$$

(7)

being $U_z$ and $U_{\theta }$ the axial and tangential velocities, $\varOmega$ the angular velocity and r the radius varying along the blade. On the other hand, the angle of attack is expressed as

$$\begin{aligned} \alpha =\varPhi -\gamma , \end{aligned}$$

(8)

where $\varPhi$ is the angle between the relative velocity and the rotor plane and, $\gamma$ is the pitch angle, as observed in Fig. 17.

$$\begin{aligned} \varPhi =tan^{-1}\left( \frac{U_z}{\varOmega r-U_{\theta }}\right) . \end{aligned}$$

(9)

5.:

Projection of hydrodynamic forces (2D) to the flow field (3D): computing the field flow, $f_i$, through the convolution integral of bi-dimensional forces and the kernel regularization function, $\eta _{\epsilon }$, such as

$$\begin{aligned} f_i({\textbf {x}})= \sum _{k=1}^{B} \int _{0}^{R} F \,{\textbf {f}}_{2D}(r)\cdot {\textbf {e}}_i \, \eta _{\epsilon } (|{\textbf {x}}- r {\textbf {e}}_k|) dr, \end{aligned}$$

(10)

where ${\textbf {e}}_k$ in the unit vector in the blade direction k, $|{\textbf {x}}- r {\textbf {e}}_k|$ is the distance between the mesh point and the actuator line point. The regularization function $\eta _{\epsilon }$ is defined as

$$\begin{aligned} \eta _{\epsilon }(r)=\frac{1}{\epsilon ^3 \pi ^{3/2}}exp[-(r/\epsilon )^2]. \end{aligned}$$

(11)

The parameter $\epsilon$ represents the reached kernel function and it is established as $\varDelta x \ge 2 \epsilon$ [22, 46, 47]. Based on the mesh generated, the $\epsilon$ value is defined as $\epsilon =2\varDelta x=0,06$.

Finally, to highlight that the function F in Eq. 10 is the correction factor to tip blade effects developed by [48] such as

$$\begin{aligned} F=\frac{2}{\pi }cos^{-1}\left[ exp \left( -g \frac{B(R-r)}{2 r sin \varPhi }\right) \right] , \end{aligned}$$

(12)

the parameter B refers to number of blades and the g coefficient depends of blade numbers, TSR value, chord distribution, pitch angle, etc... For simplicity, it established g to be only dependent on the blade numbers and the TSR [48], in the form

$$\begin{aligned} g=exp \left( -c1 \left( B \varOmega R/U_{\infty } - c2\right) \right) , \end{aligned}$$

(13)

being $c_1$ e $c_2$ coefficients estimated experimentally.

$$\begin{aligned} g=exp \left( -0.125 \left( B \varOmega R/U_{\infty } - 21\right) \right) + 0.1. \end{aligned}$$

(14)

6.:

Navier–Stokes equations solving: the flow is solved inserting the field force $f_i$ in Navier–Stokes equations,

$$\begin{aligned} \frac{\partial {u}_i}{\partial t} + {u}_j \frac{\partial {u}_i}{\partial {x}_j} = - \frac{1}{\rho } \frac{\partial {p}}{\partial {x}_i}+\nu \frac{\partial ^2 {u}_i}{\partial {x}_j \partial {x}_j}+f_i . \end{aligned}$$

(15)

After the velocity field is computed in the new time step, new angles of attack and relative velocities are defined turning to step 4, in an iterative process, where forces are computed and projected in the flow field again.

Fig. 16

Lift ($C_L$) and drag ($C_D$) coefficient values of NACA4415 hydrofoil at various angles of attack ($\alpha$) from XFOIL software [45]

Fig. 17

Hydrofoil cross section illustrating the lift and drag hydrodynamic forces and the velocities triangle